Can a p-adic ball cover a p-adic ball?

Question

Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t.

A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$ satisfy the following property.

1. $F(\mathbb{Z}_p^n)\subset B((0,...,0),1)$ where $B((0,...,0),1)$ is open ball centered at the origin with radius $1$

2. $F$ induces an isomorphism onto $B((0,...,0),1)$ on each open set $B((i_1,...,i_n),1)$ where each $i_1,...,i_n$ ranges from $0$ to $p-1$.

Note that $\mathbb{Z}_p^n=\bigcup_{i_1,...,i_n}B((i_1,...,i_n),1)$

Answer 1

We can take $f_j = x_j^p -x_j$ for each $j$ from $1,\dots n$. The equation $x_j^p - x_j = y_j$ has a unique solution $x_j \in \mathbb Z_p$ congruent to $i$ mod $p$ for each $y_j \in p \mathbb Z_p$ by Hensel's lemma. This implies the map $B((i_1,\dots,i_n),1) \to B( (0,\dots, 0),1) $ defined by $f_1,\dots, f_n$ is bijective. It is continuous, and it is easy to see the inverse provided by Hensel's lemma is continuous as well. Thus it satisfies your desiderata (assuming a topological isomorphism was meant).